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The spectral determinant det(H-εI) of the Azbel-Hofstadter Hamiltonian H is related to Onsager's partition function of the 2D Ising model for any value of magnetic flux Φ0ex0ex=0ex0ex2πP/Q through an elementary cell, where P and Q are coprime integers. The band edges of H correspond to the critical temperature of the Ising model; the spectral determinant at these (and other points defined in a certain similar way) is independent of P. A connection of the mean of Lyapunov exponents to the asymptotic (large Q) bandwidth is indicated.
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Igor Krasovsky (2000) studied this question.
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